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On the isomorphic reduction of an invariant associated with a Lie pseudogroup. (English) Zbl 0602.58057
The paper is devoted to the problem of essential invariants of Lie-Cartan pseudogroups in a slightly generalized setting. In more detail, let $$\pi$$ : $$P\to M$$ be a fibered manifold, $$u^ 1,...,u^ n$$ be functions on P, and $$\omega^ 1,...,\omega^ r$$ be 1-forms on P. We consider the pseudogroup $${\mathfrak G}(P)$$ on P consisting of all local mappings f that satisfy $$f^*u^ i\equiv u^ i$$ and $$f^*\omega^ j\equiv \omega^ j$$, and the pseudogroup $${\mathfrak G}(M)$$ on M consisting of all mappings g which are projections of the above f, i.e., $$g\circ \pi =\pi \circ f$$ with certain $$f\in {\mathfrak G}(P)$$. The problem is whether the restriction of the data $$u^ 2,...,u^ n$$, $$\omega^ 1,...,\omega^ r$$ on a level set $$u^ 1=const$$ leads to the relevant restriction of the pseudogroup with the additional property that the projection $${\mathfrak G}(M)$$ is the same as before. Assumptions on the structure equations, involutiveness, and specifying of all invariants are not needed; this is the essence of the mentioned generalization according to the common approach.
Reviewer: J.Chrastina
MSC:
 58H05 Pseudogroups and differentiable groupoids